Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, t\neq 0$. $\dfrac{{(a^{3}t^{5})^{-3}}}{{(a^{-3}t^{-2})^{-5}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{3}t^{5})^{-3} = (a^{3})^{-3}(t^{5})^{-3}}$ On the left, we have ${a^{3}}$ to the exponent ${-3}$ . Now ${3 \times -3 = -9}$ , so ${(a^{3})^{-3} = a^{-9}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{3}t^{5})^{-3}}}{{(a^{-3}t^{-2})^{-5}}} = \dfrac{{a^{-9}t^{-15}}}{{a^{15}t^{10}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-9}t^{-15}}}{{a^{15}t^{10}}} = \dfrac{{a^{-9}}}{{a^{15}}} \cdot \dfrac{{t^{-15}}}{{t^{10}}} = a^{{-9} - {15}} \cdot t^{{-15} - {10}} = a^{-24}t^{-25}$